AN EXAMPLE FOR THE USE OF BITWISE OPERATIONS IN PROGRAMMING

Transcription

1 AN EXAMPLE FOR THE USE OF BITWISE OPERATIONS IN PROGRAMMING Krasimir Yakov Yordzhev This piece of work presets a meaigful example for the advatages of usig bitwise operatios for creatig effective algorithms i programmig. A task coected with mathematical modelig i weavig idustry is examied ad computed MSC: 68N15, 68R05, 05A18 Key words: C/C++ programmig laguage, bitwise operatio, biary matrix, equivalece relatio, factor set, weaver structure, biary system, lexicographic order 1. Itroductio. The use of bitwise operatios is a powerful method used i C/C++ programmig laguages. Ufortuately i widespread books o this topic there is icomplete or o descriptio for the work of the bitwise operatios [2,4,5,9,11]. The aim of this article is to correct this lapse to a certai extet ad preset a meaigful example of a programmig task, where the use of bitwise operatios is appropriate i order to facilitate the work ad to icrease the effectiveess of the respective algorithm. O the other had the algorithm specified here could have a good practical applicatio for computig a kow combiatorial task coected with the classificatio of the various textile structures. 2. Task formulatio. Let us deote by B the set of all biary matrices, i.е. matrices composed by rows ad colums, all elemets of which are either 0 or 1. It s a well-kow fact that the umber of 2 all matrices of B is equal to 2. Let A, B B. We will say, that A ad B are equivalet ad we will write A ~ B, if B is obtaied from A as a result of sequetial cyclic move of the last row or colum at a first place. It s easy to see that the so described relatio is a equivalece relatio. So this way we come to a formulatio of the followig programmig task: Task 1. Write a program that with assiged positive iteger returs oe represetative of each equivalece class i B cocerig the above metioed equivalece relatio. As a result from the solutio of task 1 we will also compute a combiatorial task to fid the umber of all equivalece classes i B regardig the equivalece relatio ~, i.e. for fidig the cardial umber of the factor set B / ~. This task is applicable i waverig idustry. With the help of the elemets of B the various threads iterweavig of a certai weaver structure could be coded, ad with this codig by usig two equivalet matrices the weavig of oe ad the same fabric is coded, because of cyclic recurrece of the repetitio of iterweavig [6,8]. From a practical poit of view just matrices with at least oe 0 ad at least oe 1 i each row ad each colum have meaig. Let s mark with Q the set of all matrices of that kid, Q B. The ext task which we are goig to compute is a bit more difficult versio of task 1.

2 Task 2. Write a program that with assiged positive iteger returs oe represetative of each equivalece class i Q cocerig the above metioed equivalece relatio. 3. Bitwise operatios i C/C++. Bitwise operatios ca be applied for iteger data type oly, i.e. they caot be used for float ad double types. For the defiitio of the bitwise operatios i C/C++ ad some of their elemetary applicatios could be see, for example, i [1,3,7,10]. We assume as usually that bits umberig i variables starts from right to left, ad that the umber of the very right oe is 0. Let x, y ad z are iteger variables of oe type, for which w bits are eeded. Let x ad y are iitialized ad let the z x y assigmet is made, where is oe of the operators & (bitwise AND), (bitwise iclusive OR) or ^ (bitwise exclusive OR). For each i 0,1,, w 1 the ew cotets of the i bit i z will be as it is preseted i the followig table: The i bit of x The i bit of y The i bit of x & y The i bit of x y Thei bit of x^ y z ~ x I case that, if the i bit of x is 0, the the i bit of z becomes 1, ad if the i bit of x is 1, the the i bit of z becomes 0, i 0,1,, w 1. I case that k is a oegative iteger, the the statemet z x k; (bitwise shift left) will write i the ( i k) bit of z the value of the k bit of x, wherei 0,1,, w k 1, ad the very right k bits of z will be filled by zeroes. This operatio is equivalet to a multiplicatio of k x by 2. The statemet z x k works the similar way (bitwise shift right). But we must be careful here, as i various programmig eviromets (see for example i [7]) this operatio has differet iterpretatios somewhere k bits of z from the very left place are compulsory filled by 0 (logical displacemet), ad elsewhere the very left k bits of z are filled with the value from the very left (sig) bit; i.e. if the umber is egative, the the fillig will be with 1 (arithmetic displacemet). Therefore it s recommeded to use usiged type of variables (if the opposite is ot ecessary) while workig with bitwise operatios. Directly form the defiitio of the operatio bitwise shift left follows k the effectiveess of the followig fuctio computig 2, where k is a oegative iteger: usiged it Power2(usiged it k) { retur 1<<k; To compute the value of the i bit of a iteger variable x we ca use the fuctio:

3 it BitValue(it x, usiged it i) { if ( (x & (1<<i) ) == 0 ) retur 0; else retur 1; Bitwise operatios are left associative. The priority of operatios i descedig order is as follows: bitwise complemet ~; arithmetic operatios * (multiply), / (divide), % (remaider or modulus); arithmetic operatios + (biary plus or add) - (biary mius or subtract); the bitwise operatios << ad >>; relatioal operatios <, >, <=, >=, ==,!=; bitwise operatios &,^ ad ; logical operatios && ad. 4. Algorithm realizatio. Each biary matrix A ca be coded with the help of vector (array) of oegative itegers v ( v0, v1,, v 1), where 0 v i 2 1 for each i : 0 i 1. Oe-to-oe correspodece is realized through biary presetatio of atural umbers, i.e. the i row of the matrix A is v i i biary system. The row i of A will be completely il if ad oly if v i 0 ; ad all elemets of the i row of A will be equal to 1 if ad oly if v 2 i 1. I other words, it s a ecessary ad sufficiet coditio for each i 0,1,, 1 to be realized 1 v i 2 2, i order to obtai at least oe 0 ad at least oe 1 i each row. I order to obtai at least oe 0 i each colum of the matrix A, it is ecessary ad sufficiet that the bitwise AND of all umbers, represetig the rows of A to be equal to 0. I order to obtai at least oe 1 i each colum of the matrix A it is ecessary ad sufficiet that the bitwise iclusive OR of all umbers, represetig the rows of A to be equal to 2 1, i.e. to be equal to a umber which is writte i biary system with exact umber of 1 ad ot eve oe 0. Thus we obtai the followig fuctio, which checks whether the array of itegers v v, v,, v represets a matrix of Q, or ot ) ( it IsQ(usiged it v[], usiged it ) { // Returs 1, if with v a matrix i Q is coded // Returs 0, otherwise for (it i=0; i <= -1; i++) if (v[i]<1 v[i] > (1<<)-2) retur 0; it x,y; x = (1<<) -1; y=0; for (it j=0; j <= -1; j++) { x = x & v[j]; y = y v[j]; if (x!= 0) retur 0; if (y!= (1<<)-1) retur 0; retur 1;

4 Let x be a iteger, for which we are certai that it belogs to the iterval 0 x 2 1, i.e. there s o eed of more tha digits 0 or 1 for its biary code. The to preset x i biary system (see the fuctio BitValue described i the previous sectio), writte with the aid of exactly digits 0 or 1 ad evetually with a certai umber of isigificat zeroes at the begiig, we ca use the followig fuctio: void BiPr(it x, usiged it k) { it z; for (it i = k-1; i >= 0; i--) { z = x & (1<<i); if (z == 0) cout<< 0 ; else cout<< 1 ; cout<< \ ; Let us examie the set V {( v0, v1,, v 1) 0 v 2 1, i 0,1,, 1. i All elemets of V ca be sorted i ascedig lexicographic order. The essece of the proposed by us algorithm is to obtai sequetially all elemets of V i the same icreasig order from the smallest oe to the biggest oe ad right after obtaiig them to check whether this elemet is miimal accordig to the lexicography order i he class of equivalece. At last we will separate just the miimal i their class of equivalecy elemets ad they will be the oly represetatives of each equivalet class i the sets ad Q (which was required i Tasks 1 ad 2). For this purpose we B will desig fuctio IsMi, which will retur 1, if the iput argumet is miimal i the class of equivalecy to which it belogs to, ad 0 otherwise. But before that we eed the followig auxiliary fuctio CicleMove, which from assiged oegative itegers x ad retur a umber, which is obtaied from x by movig all bits with oe to the right, begiig with the movig of the very right bit to the place of the bit 1. I this case we will be helped by the bitwise operatios. usiged it CicleMove(usiged it x,usiged it ) { usiged it b0 = x & 1; // Record the value //of the very right bit of x x=x & ((1<<)-1); // Replaces all bits to the //left from the o with umber (-1) with 0 retur (x >> 1) (b0 << -1); The followig auxiliary fuctio will also be useful for the computig of the mai task: It IsLess(usiged it u[],usiged it v[],it ) { // Retur 1, if accordig to lexicographic order //u[0] u[1] u[-1] < v[0] v[1] v[-1] // Retur 0, otherwise

5 it i = 0; while ((u[i] == v[i]) && (i<-1)) i++; if (u[i] < v[i]) retur 1; else retur 0; The above metioed fuctio IsMi could look as follows: it IsMi(usiged it v[], usiged it ) { // Retur 1, if accordig to lexicographic order //v[0] v[1] v[-1]is miimal i its class of // equivalecy // Retur 0, otherwise usiged it u[32], v1[32]; for (it i = 0; i <= -1; i++) v1[i] = v[i]; for (it j = 0; j <= -1; j++) { for (it i = 1; i <= -1; i++) { for (it s = 0; s <= -1; s++) { it s1 = (s+i) % ; u[s] = v1[s1]; if (IsLess(u,v,) ) retur 0; for (it i=0; i <= -1; i++) v1[i]=ciclemove(v1[i],); retur 1; Takig the advatages of the above described fuctios we propose the followig computig of tasks 1 ad 2 (for =4, for example). I order to be brief here we will ot prit all the elemets obtaied, ad we will obtai their umber oly. For the hard-copy itself for each row of ay of the obtaied matrices we ca take advatage of the above described procedure BiPr ad after orgaizig of a cycle by the umber of the row to prit the whole matrix as well.. it mai() { cost it =4; it i; usiged log it NB = 0; usiged log it NQ = 0; // Number of elemets //i B // Number of elemets //i Q usiged log it NBEq = 0; // Number of the //classes of equivalecy i B usiged log it NQEq = 0; // Number of the //classes of equivalecy i Q usiged it v[]; it r=(1<<)-1; for (i = 0; i<; i++) v[i]=0; do { i=-1; for (it k=0; k<=r; k++) { v[i]=k; NB++;

6 if (IsQ(v,) ) NQ++; if (IsMi(v,) ) { NBEq++; if (IsQ(v,)) NQEq++; while (v[i]==r) i--; if (i>=0) { v[i]++; for (it k=-1; k>i; k--) { v[k]=0; while ( i>=0 ); cout<< Number of elemets i B <<NB<< \ ; cout<< Number of elemets i Q <<NQ<< \ ; cout<< Number of the classes of equivalecy i B <<NBEq<< \ ; cout<< Number of the classes of equivalecy i Q <<NQEq<< \ ; retur 0; The results from the above described program for some values of ca be summarized i the followig table B Q B Q / ~ / ~ REFERENCES [1] S. R. DAVIS C++ for dummies. IDG Books Worldwide, [2] C. S. HORSTMANN Computig cocepts with C++ essetials. Joh Wiley & Sos,1999. [3] B. W. KERNIGAN, D. M RITCHIE The C programmig Laguage. AT&T Bell Laboratories, [4] П. АЗЪЛОВ Информатика Езикът С++ в примери и задачи за 9-10 клас. София, Просвета, [5] П. АЗЪЛОВ Обектно ориентирано програмиране Структури от данни и STL. София, Сиела, [6] Г. И. БОРЗУНОВ Шерстяная промишленост-обзорная информащия. Москва, ЦНИИ ИТЭИЛП, 3, (1983). [7] С. В. ГЛУШАКОВ, А. В. КОВАЛЬ, С. В. СМИРНОВ Язык программирования C++. Харьков, Фолио, [8] К. Я. ЙОРДЖЕВ, И. В. СТАТУЛОВ Математическо моделиране и количествена оценка на първичните тъкачни сплитки. Текстил и облекло, 10, (1999), [9] Х. КРУШКОВ Практическо ръководство по програмиране на С++. Пловдив, Макрос,2006.

7 [10] Е. Л. РОМАНОВ Практикум по программированию на C++. Санкт Петербург, БХВ, [11] М. ТОДОРОВА Програмиране на С++. Част І, част ІІ, София, Сиела, Krasimir Yakov Yordzhev South-West Uiversity N. Rilsky 2700 Blagoevgrad, Bulgaria